Statistical Modelling of COVID-19 Outbreak in Italy

24 Apr 2020



Nonlinear growth models

Nonlinear growth models represent an instance of nonlinear regression models, a class of models taking the general form \[ y = \mu(x, \theta) + \epsilon, \] where \(\mu(x, \theta)\) is the mean function which depends on a possibly vector-valued parameter \(\theta\), and a possibly vector-valued predictor \(x\). The stochastic component \(\epsilon\) represents the error with mean zero and constant variance. Usually, a Gaussian distribution is also assumed for the error term.

By defining the mean function \(\mu(x, \theta)\) we may obtain several different models, all characterized by the fact that parameters \(\theta\) enter in a nonlinear way into the equation. Parameters are usually estimated by nonlinear least squares which aims at minimizing the residual sum of squares.

Exponential

\[ \mu(x) = \theta_1 \exp\{\theta_2 x\} \] where \(\theta_1\) is the value at the origin (i.e. \(\mu(x=0)\)), and \(\theta_2\) represents the (constant) relative ratio of change (i.e. \(\frac{d\mu(x)}{dx }\frac{1}{\mu(x)} = \theta_2\)). Thus, the model describes an increasing (exponential growth if \(\theta_2 > 0\)) or decreasing (exponential decay if \(\theta_2 < 0\)) trend with constant relative rate.

Logistic

\[ \mu(x) = \frac{\theta_1}{1+\exp\{(\theta_2 - x)/\theta_3\}} \] where \(\theta_1\) is the upper horizontal asymptote, \(\theta_2\) represents the x-value at the inflection point of the symmetric growth curve, and \(\theta_3\) represents a scale parameter (and \(1/\theta_3\) is the growth-rate parameter that controls how quickly the curve approaches the upper asymptote).

Gompertz

\[ \mu(x) = \theta_1 \exp\{-\theta_2 \theta_3^x\} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the value of the function at \(x = 0\) (displacement along the x-axis), and \(\theta_3\) represents a scale parameter.

The difference between the logistic and Gompertz functions is that the latter is not symmetric around the inflection point.

Richards

\[ \mu(x) = \theta_1 (1 - \exp\{-\theta_2 x\})^{\theta_3} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the rate of growth, and \(\theta_3\) in part determines the point of inflection on the y-axis.

Data

Dipartimento della Protezione Civile: COVID-19 Italia - Monitoraggio della situazione http://arcg.is/C1unv

Source: https://github.com/pcm-dpc/COVID-19

## ## Avvisi
## 
## ```diff
## - 23/04/2020: dati Regione Lazio parziali (casi testati non completi)
## - 23/04/2020: dati Regione Campania parziali (casi testati non aggiornati)
## - 21/04/2020: dati Regione Lombardia parziali (casi testati non aggiornati)
## - 20/04/2020: dati Regione Lombardia ricalcolati (ricalcolo di casi testati - eliminazione duplicati)
## - 15/04/2020: dati Regione Friuli Venezia Giulia ricalcolati (ricalcolo di isolamento domiciliare e dimessi/guariti)
## - 12/04/2020: dati P.A. Bolzano ricalcolati (ricalcolo dati guariti -110 rispetto a ieri)
## - 10/04/2020: dati Regione Molise parziali (dato tamponi non aggiornato)
## - 29/03/2020: dati Regione Emilia-Romagna parziali (dato tamponi non aggiornato)
## - 26/03/2020: dati Regione Piemonte parziali (-50 deceduti - comunicazione tardiva)
## - 18/03/2020: dati Regione Campania non pervenuti
## - 18/03/2020: dati Provincia di Parma non pervenuti
## - 17/03/2020: dati Provincia di Rimini non aggiornati
## - 16/03/2020: dati P.A. Trento e Puglia non pervenuti
## - 11/03/2020: dati Regione Abruzzo non pervenuti
## - 10/03/2020: dati Regione Lombardia parziali
## - 07/03/2020: dati Brescia +300 esiti positivi
## ```
url = "https://raw.githubusercontent.com/pcm-dpc/COVID-19/master/dati-andamento-nazionale/dpc-covid19-ita-andamento-nazionale.csv"
COVID19 <- read.csv(file = url, stringsAsFactors = FALSE)
COVID19$data <- as.Date(COVID19$data)
# DT::datatable(COVID19)


Modelling total infected

# create data for analysis
data = data.frame(date = COVID19$data,
                  y = COVID19$totale_casi,
                                    dy = reldiff(COVID19$totale_casi))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))

Estimation

Exponential

mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
## 
## Formula: y ~ exponential(x, th1, th2)
## 
## Parameters:
##         Estimate   Std. Error t value         Pr(>|t|)    
## th1 17872.452879  2060.250431   8.675 0.00000000000399 ***
## th2     0.041860     0.002256  18.558          < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19950 on 59 degrees of freedom
## 
## Number of iterations to convergence: 12 
## Achieved convergence tolerance: 0.000008476

Logistic

mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
## 
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
## 
## Parameters:
##         Estimate  Std. Error t value Pr(>|t|)    
## Asym 191623.3802   2473.2931   77.48   <2e-16 ***
## xmid     35.6320      0.3365  105.88   <2e-16 ***
## scal      8.1347      0.2312   35.19   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4459 on 58 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.0000007395

Gompertz

mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
#            control = nls.control(maxiter = 1000))
summary(mod3)
## 
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
## 
## Parameters:
##           Estimate    Std. Error t value Pr(>|t|)    
## Asym 220385.823585   1817.674575  121.25   <2e-16 ***
## b2        8.560171      0.192674   44.43   <2e-16 ***
## b3        0.936063      0.000883 1060.06   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1612 on 58 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.00000002705

Richards

richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss  <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2) 
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss, 
               y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
           # trace = TRUE, algorithm = "plinear", 
           control = nls.control(maxiter = 1000, tol = 0.1))
# algorithm is not converging... 
summary(mod4)
## 
## Formula: y ~ richards(x, th1, th2, th3)
## 
## Parameters:
##          Estimate    Std. Error t value Pr(>|t|)    
## th1 230858.329274   2268.936708  101.75   <2e-16 ***
## th2      0.056716      0.001052   53.94   <2e-16 ***
## th3      6.033078      0.167957   35.92   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1438 on 58 degrees of freedom
## 
## Number of iterations to convergence: 2 
## Achieved convergence tolerance: 0.03223
# library(nlmrt)
# mod4 = nlxb(y ~ th1*(1 - exp(-th2*x))^th3, 
#             data = data, start = start, trace = TRUE)

Models comparison

models = list("Exponential model" = mod1, 
              "Logistic model" = mod2, 
              "Gompertz model" = mod3,
              "Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
                 df = sapply(models, function(m) attr(logLik(m), "df")),
                 Rsquare = sapply(models, function(m) 
                                  cor(data$y, fitted(m))^2),
                 AIC = sapply(models, AIC),
                 AICc = sapply(models, AICc),
                 BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
                 cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)
loglik df Rsquare AIC AICc BIC
Exponential model -689.4914 3 0.9244200 1384.983 1385.404 1391.315
Logistic model -597.5738 4 0.9963899 1203.148 1203.862 1211.591
Gompertz model -535.5003 4 0.9994553 1079.001 1079.715 1087.444
Richards model -528.5392 4 0.9995814 1065.078 1065.793 1073.522 *** 
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(aes(y = fitted(mod1), color = "Exponential")) +
  geom_line(aes(y = fitted(mod2), color = "Logistic")) +
  geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
  geom_line(aes(y = fitted(mod4), color = "Richards")) +
  labs(x = "", y = "Infected", color = "Model") +
  scale_color_manual(values = cols) +
  scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 10000),
                     minor_breaks = seq(0, coef(mod2)[1], by = 5000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

last_plot() +
  scale_y_continuous(trans = "log10", limits = c(100,NA)) +
  labs(y = "Infected (log10 scale)")

Predictions

Point estimates

df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
               fit1 = predict(mod1, newdata = df),
               fit2 = predict(mod2, newdata = df),
               fit3 = predict(mod3, newdata = df),
               fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,c("fit2", "fit3")]))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
  geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
  geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
  geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
  coord_cartesian(ylim = ylim) +
  labs(x = "", y = "Infected", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 10000),
                     minor_breaks = seq(0, max(ylim), by = 5000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Prediction intervals

# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))

pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]

pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]

pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]

pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]

# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
             subset(pred2, x == max(data$x)+1, select = 2:5),
             subset(pred3, x == max(data$x)+1, select = 2:5),
             subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
##           date    fit    lwr    upr
## 62  2020-04-25 239500 185977 293272
## 621 2020-04-25 184411 173807 193392
## 622 2020-04-25 191141 187163 194837
## 623 2020-04-25 192455 188832 196135

ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
  geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
  geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
  geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
  geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
  geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
  geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
  geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
  coord_cartesian(ylim = c(0, max(ylim))) +
  labs(x = "", y = "Infected", color = "Model") +
  scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 10000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Modelling total deceased

# create data for analysis
data = data.frame(date = COVID19$data,
                  y = COVID19$deceduti,
                                    dy = reldiff(COVID19$deceduti))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))

Estimation

Exponential

mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
## 
## Formula: y ~ exponential(x, th1, th2)
## 
## Parameters:
##        Estimate  Std. Error t value      Pr(>|t|)    
## th1 1727.352332  220.861864   7.821 0.00000000011 ***
## th2    0.047263    0.002455  19.249       < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2521 on 59 degrees of freedom
## 
## Number of iterations to convergence: 13 
## Achieved convergence tolerance: 0.000004033

Logistic

mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
## 
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
## 
## Parameters:
##        Estimate Std. Error t value Pr(>|t|)    
## Asym 25985.7246   348.5311   74.56   <2e-16 ***
## xmid    38.4792     0.3194  120.47   <2e-16 ***
## scal     7.6139     0.2124   35.85   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 560.3 on 58 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000001155

Gompertz

mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# manually set starting values
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start, 
#            control = nls.control(maxiter = 10000))
summary(mod3)
## 
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
## 
## Parameters:
##           Estimate    Std. Error t value Pr(>|t|)    
## Asym 30445.5879075   262.8273593  115.84   <2e-16 ***
## b2      11.7580795     0.3000062   39.19   <2e-16 ***
## b3       0.9335651     0.0008958 1042.19   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 196 on 58 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000002779

Richards

richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss  <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2) 
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss, 
               y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
           # trace = TRUE, algorithm = "port", 
           control = nls.control(maxiter = 1000))
summary(mod4)
## 
## Formula: y ~ richards(x, th1, th2, th3)
## 
## Parameters:
##         Estimate   Std. Error t value Pr(>|t|)    
## th1 31541.649850   306.790026  102.81   <2e-16 ***
## th2     0.061748     0.001045   59.11   <2e-16 ***
## th3     8.912991     0.267317   33.34   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 177.9 on 58 degrees of freedom
## 
## Number of iterations to convergence: 5 
## Achieved convergence tolerance: 0.000002185

Models comparison

models = list("Exponential model" = mod1, 
              "Logistic model" = mod2, 
              "Gompertz model" = mod3,
              "Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
                 df = sapply(models, function(m) attr(logLik(m), "df")),
                 Rsquare = sapply(models, function(m) 
                                  cor(data$y, fitted(m))^2),
                 AIC = sapply(models, AIC),
                 AICc = sapply(models, AICc),
                 BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
                 cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)
loglik df Rsquare AIC AICc BIC
Exponential model -563.3106 3 0.9352170 1132.6212 1133.0423 1138.9539
Logistic model -471.0573 4 0.9969086 950.1146 950.8289 958.5581
Gompertz model -406.9786 4 0.9995588 821.9571 822.6714 830.4006
Richards model -401.0869 4 0.9996392 810.1739 810.8882 818.6174 *** 
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(aes(y = fitted(mod1), color = "Exponential")) +
  geom_line(aes(y = fitted(mod2), color = "Logistic")) +
  geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
  geom_line(aes(y = fitted(mod4), color = "Richards")) +
  labs(x = "", y = "Deceased", color = "Model") +
  scale_color_manual(values = cols) +
  scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
                     minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

last_plot() +
  scale_y_continuous(trans = "log10", limits = c(10,NA)) +
  labs(y = "Deceased (log10 scale)")

Predictions

Point estimates

df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
               fit1 = predict(mod1, newdata = df),
               fit2 = predict(mod2, newdata = df),
               fit3 = predict(mod3, newdata = df),
               fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
  geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
  geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
  geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
  coord_cartesian(ylim = ylim) +
  labs(x = "", y = "Deceased", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
                     minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Prediction intervals

# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))

pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]

pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]

pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]

pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]

# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
             subset(pred2, x == max(data$x)+1, select = 2:5),
             subset(pred3, x == max(data$x)+1, select = 2:5),
             subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
##           date   fit   lwr   upr
## 62  2020-04-25 32359 25099 39751
## 621 2020-04-25 24854 23452 25849
## 622 2020-04-25 25797 25291 26238
## 623 2020-04-25 25929 25476 26410

ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
  geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
  geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
  geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
  geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
  geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
  geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
  geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
  coord_cartesian(ylim = c(0, max(ylim))) +
  labs(x = "", y = "Deceased", color = "Model") +
  scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Modelling recovered

# create data for analysis
data = data.frame(date = COVID19$data,
                  y = COVID19$dimessi_guariti,
                                    dy = reldiff(COVID19$dimessi_guariti))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))

Estimation

Exponential

mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
## 
## Formula: y ~ exponential(x, th1, th2)
## 
## Parameters:
##        Estimate  Std. Error t value Pr(>|t|)    
## th1 1486.869383  124.399277   11.95   <2e-16 ***
## th2    0.061749    0.001549   39.87   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2327 on 59 degrees of freedom
## 
## Number of iterations to convergence: 11 
## Achieved convergence tolerance: 0.000005316

Logistic

mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
## 
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
## 
## Parameters:
##        Estimate Std. Error t value Pr(>|t|)    
## Asym 84929.1742  3832.9294   22.16   <2e-16 ***
## xmid    53.2351     0.9912   53.71   <2e-16 ***
## scal    10.2848     0.3097   33.21   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1053 on 58 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000002722

Gompertz

mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
summary(mod3)
## 
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
## 
## Parameters:
##           Estimate    Std. Error t value Pr(>|t|)    
## Asym 207779.444999  16625.409278    12.5   <2e-16 ***
## b2        8.069375      0.142832    56.5   <2e-16 ***
## b3        0.970068      0.001206   804.2   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 598.9 on 58 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000002003

Richards

richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss  <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2) 
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss, 
               y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
           # trace = TRUE, # algorithm = "port", 
           control = nls.control(maxiter = 1000))
summary(mod4)
## 
## Formula: y ~ richards(x, th1, th2, th3)
## 
## Parameters:
##          Estimate    Std. Error t value    Pr(>|t|)    
## th1 833096.216313 258251.105513   3.226     0.00207 ** 
## th2      0.010408      0.001752   5.940 0.000000171 ***
## th3      3.503522      0.154415  22.689     < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 475.3 on 58 degrees of freedom
## 
## Number of iterations to convergence: 36 
## Achieved convergence tolerance: 0.000003554

Models comparison

models = list("Exponential model" = mod1, 
              "Logistic model" = mod2, 
              "Gompertz model" = mod3,
              "Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
                 df = sapply(models, function(m) attr(logLik(m), "df")),
                 Rsquare = sapply(models, function(m) 
                                  cor(data$y, fitted(m))^2),
                 AIC = sapply(models, AIC),
                 AICc = sapply(models, AICc),
                 BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
                 cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)
loglik df Rsquare AIC AICc BIC
Exponential model -558.4346 3 0.9871020 1122.8692 1123.2903 1129.2019
Logistic model -509.5352 4 0.9971410 1027.0704 1027.7847 1035.5139
Gompertz model -475.1140 4 0.9989753 958.2280 958.9422 966.6714
Richards model -461.0213 4 0.9993315 930.0426 930.7569 938.4861 *** 
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(aes(y = fitted(mod1), color = "Exponential")) +
  geom_line(aes(y = fitted(mod2), color = "Logistic")) +
  geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
  geom_line(aes(y = fitted(mod4), color = "Richards")) +
  labs(x = "", y = "Recovered", color = "Model") +
  scale_color_manual(values = cols) +
  scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
                     minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

last_plot() +
  scale_y_continuous(trans = "log10", limits = c(10,NA)) +
  labs(y = "Recovered (log10 scale)")

Predictions

Point estimates

df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
               fit1 = predict(mod1, newdata = df),
               fit2 = predict(mod2, newdata = df),
               fit3 = predict(mod3, newdata = df),
               fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() + 
  geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
  geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
  geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
  geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
  coord_cartesian(ylim = ylim) +
  labs(x = "", y = "Recovered", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
                     minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Prediction intervals

# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))

pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]

pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]

pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]

pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]

# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
             subset(pred2, x == max(data$x)+1, select = 2:5),
             subset(pred3, x == max(data$x)+1, select = 2:5),
             subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
##           date   fit   lwr   upr
## 62  2020-04-25 68381 61638 75373
## 621 2020-04-25 59538 56248 61913
## 622 2020-04-25 60961 59292 62224
## 623 2020-04-25 61600 60316 62593

ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
  geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
  geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
  geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
  geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
  geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
  geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
  geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
  coord_cartesian(ylim = c(0, max(ylim))) +
  labs(x = "", y = "Recovered", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 5000),
                     minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Description of evolution

Positive cases and administered swabs

df = data.frame(date = COVID19$data,
                positives = c(NA, diff(COVID19$totale_casi)),
                swabs = c(NA, diff(COVID19$tamponi)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
# df$y = df$positives/df$swabs
df$y = df$positives/c(NA, zoo::rollmean(df$swabs, 2))
df = subset(df, swabs > 50)
# DT::datatable(df[,-4], )
ggplot(df, aes(x = date)) + 
  geom_point(aes(y = swabs, color = "swabs"), pch = 19) +
  geom_line(aes(y = swabs, color = "swabs")) +
  geom_point(aes(y = positives, color = "positives"), pch = 0) +
  geom_line(aes(y = positives, color = "positives")) +
  labs(x = "", y = "Number of cases", color = "") +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = palette()[c(2,1)]) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

ggplot(df, aes(x = date, y = y)) + 
  geom_smooth(method = "loess", se = TRUE, col = "black") +
  geom_point(col=palette()[4]) + 
  geom_line(size = 0.5, col=palette()[4]) +
  labs(x = "", y = "% positives among admnistered swabs (two-day rolling mean)") +
  scale_y_continuous(labels = scales::percent_format(),
                     breaks = seq(0, 0.5, by = 0.05)) +
  coord_cartesian(ylim = c(0,max(df$y, na.rm = TRUE))) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Hospitalized and ICU patients

df = data.frame(date = COVID19$data,
                hospital = c(NA, diff(COVID19$totale_ospedalizzati)),
                icu = c(NA, diff(COVID19$terapia_intensiva)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
ggplot(df, aes(x = date, y = hospital)) + 
  geom_smooth(method = "loess", se = TRUE, col = "black") +
  geom_point(col = "orange") + 
  geom_line(size = 0.5, col = "orange") +
  labs(x = "", y = "Change hospitalized patients") +
  coord_cartesian(ylim = range(df$hospital, na.rm = TRUE)) +
  scale_y_continuous(minor_breaks = seq(min(df$hospital, na.rm = TRUE),
                                        max(df$hospital, na.rm = TRUE), 
                                        by = 100)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

ggplot(df, aes(x = date, y = icu)) + 
  geom_smooth(method = "loess", se = TRUE, col = "black") +
  geom_point(col = "red2") + 
  geom_line(size = 0.5, col = "red2") +
  labs(x = "", y = "Change ICU patients") +
  coord_cartesian(ylim = range(df$icu, na.rm = TRUE)) +
  scale_y_continuous(minor_breaks = seq(min(df$icu, na.rm = TRUE), 
                                        max(df$icu, na.rm = TRUE), 
                                        by = 10)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))